# Direct product of cyclic group of prime-square order and cyclic group of prime-square order

From Groupprops

This article is about a family of groups with a parameter that is prime. For any fixed value of the prime, we get a particular group.

View other such prime-parametrized groups

## Contents

## Definition

Let be a prime number. This group is defined in the following equivalent ways:

- It is the external direct product of two copies of the cyclic group of prime-square order, i.e., it is the group .
- It is the homocyclic group of order and exponent .

## Particular cases

Value of prime number | Corresponding group |
---|---|

2 | direct product of Z4 and Z4 |

3 | direct product of Z9 and Z9 |

5 | direct product of Z25 and Z25 |

## Arithmetic functions

## GAP implementation

### Group ID

This finite group has order p^4"p^" can not be assigned to a declared number type with value 4.

and has ID 2 among the groups of order p^4 in GAP's SmallGroup library. For context, there are groups of order p^4. It can thus be defined using GAP's SmallGroup function as:
`SmallGroup(p^4,2)`

For instance, we can use the following assignment in GAP to create the group and name it :

`gap> G := SmallGroup(p^4,2);`

Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:

`IdGroup(G) = [p^4,2]`

or just do:

`IdGroup(G)`

to have GAP output the group ID, that we can then compare to what we want.

"p^" can not be assigned to a declared number type with value 4.

### Short descriptions

Description | Functions used | Mathematical comments |
---|---|---|

DirectProduct(CyclicGroup(p^2),CyclicGroup(p^2)) |
DirectProduct and CyclicGroup |